Multinomial Logit vs. Multinomial Probit

Attribute	Multinomial Logit	Multinomial Probit
Model Type	Logit	Probit
Assumption of Error Term	Independently and identically distributed (i.i.d) Gumbel	Independently and identically distributed (i.i.d) Normal
Interpretation of Coefficients	Change in log-odds	Change in probability
Computational Complexity	Less computationally intensive	More computationally intensive
Estimation Method	Maximum Likelihood Estimation	Maximum Likelihood Estimation

Introduction

When it comes to modeling categorical outcomes with more than two categories, researchers often turn to multinomial logit and multinomial probit models. Both of these models are commonly used in fields such as economics, political science, and sociology to analyze choices made by individuals or groups. While both models have their strengths and weaknesses, understanding the differences between them can help researchers choose the most appropriate model for their specific research question.

Modeling Approach

The multinomial logit model assumes that the errors in the model are distributed according to a logistic distribution. This means that the probability of choosing a particular category is modeled as a function of the explanatory variables through a logistic transformation. On the other hand, the multinomial probit model assumes that the errors follow a normal distribution. In this model, the probability of choosing a category is modeled as a function of the explanatory variables through a cumulative normal distribution.

Interpretation of Coefficients

One of the key differences between multinomial logit and multinomial probit models lies in the interpretation of the coefficients. In the multinomial logit model, the coefficients can be interpreted as log odds ratios. This means that a one-unit increase in an explanatory variable leads to a multiplicative change in the odds of choosing a particular category. In contrast, the coefficients in the multinomial probit model cannot be directly interpreted as odds ratios. Instead, they represent the change in the probability of choosing a category associated with a one-unit increase in the explanatory variable.

Computational Considerations

Another important difference between multinomial logit and multinomial probit models is the computational complexity involved in estimating the models. The multinomial logit model can be estimated using maximum likelihood estimation, which is relatively straightforward and computationally efficient. On the other hand, the multinomial probit model requires more complex estimation techniques, such as Markov chain Monte Carlo methods, which can be computationally intensive and time-consuming.

Model Fit

When comparing the fit of multinomial logit and multinomial probit models, researchers often look at measures such as the likelihood ratio test or the Akaike Information Criterion (AIC). These measures can help determine which model provides a better fit to the data. In general, the multinomial logit model tends to have better fit statistics compared to the multinomial probit model, especially when the assumptions of the logit model are met.

Robustness

One advantage of the multinomial probit model is its robustness to violations of the independence of irrelevant alternatives (IIA) assumption. The IIA assumption states that the relative odds of choosing one category over another are independent of the presence of other categories. When this assumption is violated, the multinomial logit model may produce biased estimates. In contrast, the multinomial probit model does not rely on the IIA assumption and can provide more robust estimates in the presence of correlated alternatives.

Conclusion

In conclusion, both multinomial logit and multinomial probit models are valuable tools for analyzing categorical outcomes with multiple categories. While the multinomial logit model is easier to interpret and estimate, the multinomial probit model offers greater flexibility and robustness in certain situations. Researchers should carefully consider the assumptions and requirements of each model before choosing the most appropriate one for their research question. By understanding the differences between these two models, researchers can make informed decisions about which model best suits their needs.